A process is modeled in continuous time by a second-order-plus-time-delay transfer function with K=1,\space τ_{I}=5,\space and\space τ_{2}=3. For Δt=1, the discrete-time equivalent (with ZOH) is
G= \frac {(b_{1}+b_{2}z^{-1})z^{-N-1}}{1+ a_{1}z^{-1}+ a_{2}z^{-2}}
where a_{1}=−1.5353,\space a_{2}=0.5866,\space b_{1} =0.0280, \space b_{2} =0.0234, and N=0 (cf. Equation
y(k)=a_{1}y(k−1)+a_{2}y(k−2)+b_{1}u(k−1) +b_{2}u(k−2)
where
a_{1} = e^{−Δt/τ_{1} + e^{−Δt∕τ_{2}} }
a_{2} = -e^{−Δt/τ_{1} }e^{−Δt∕τ_{2}}
b_{1} = K(1+\frac{\tau _{a}-\tau _{1}}{\tau _{1}-\tau _{2}}e^{-\Delta t/\tau _{1}}+\frac{\tau _{2}-\tau _{a}}{\tau _{1}-\tau _{2}}e^{-\Delta t/\tau _{2}} )
b_{2} = K(e^{-\Delta t(1/\tau _{1}+1/\tau _{2})}+\frac{\tau _{a}-\tau _{1}}{\tau _{1}-\tau _{2}}e^{-\Delta t/\tau _{2}}+\frac{\tau _{2}-\tau _{a}}{\tau _{1}-\tau _{2}}e^{-\Delta t/\tau _{1}} ) )
For Dahlin’s controller with τ_{c} =Δ t = 1, plot the response for a unit change in set point at t=5 for 0≤t≤10 using Simulink.